Mathematical Modeling: Intro

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Mathematical ModelingIntro

• Patrice Koehl
• Department of Biological Sciences
• National University of Singapore

http//www.cs.ucdavis.edu/koehl/Teaching/BL5229 d
bskoehl_at_nus.edu.sg
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Science, then, and now
At the beginning, there were thoughts, and observa
tion.
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Science, then, and now

• For a long time, people thought that it would be
  enough to reason about the existing knowledge to
  explore everything there is to know.
• One single person could possess all knowledge in
  her cultural context.
• (encyclopedia of Diderot and DAlembert)
• Reasoning, and mostly passive observation were
  the main techniques in scientific research

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Science, then, and now
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Science, then, and now

• Todays experiment yields massive amounts of data
• From hypothesis-driven to exploratory data
  analysis
• - data are used to formulate
• new hypotheses
• - computers help formulate
  hypotheses
• No single person, no group has an overview of
  what is known

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Science, then, and now
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Science, then, and now

• Computer simulations developed hand-in-hand with
• the rapid growth of computers.
• A computer simulation is a computer program that
• attempts to simulate an abstract model of a
  particular
• system
• Computer simulations complement theory and
• experiments, and often integrate them
• They are becoming widesepread in Computational
• Physics, Chemistry, Mechanics, Materials, ,
  Biology

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Science, then, and now
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Mathematical Modeling

• Is often used in place of experiments when they
  are too large, too expensive, too dangerous, or
  too time consuming.
• Can be useful in what if studies e.g. to
  investigate the use of pathogens (viruses,
  bacteria) to control an insect population.
• Is a modern tool for scientific investigation.

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Mathematical Modeling
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Mathematical Modeling
Real World

• Define real world problem
• - Perform background research
• Perform experiments,
• if appropriate

Task Understand current activity and predict
future behavior
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Mathematical Modeling

• Simplification define model
• Identify and select factors to
  
• describe important aspects of
• the Real World Problem
• determine those factors
• that can be neglected.

Simplified Model
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Mathematical Modeling

• Represent mathematical model
• Express the simplified model
• in mathematical terms
• the success of a
• mathematical model depends
• on how easy it is to use and
• how accurately it predicts

Mathematical Model
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Mathematical Modeling

• Translate computational model
• Change Mathematical
• Model into a form suitable
• for computational solution
• Choice of the numerical method
• Choice of the algorithm
• Choice of the software (Matlab)

Computatonal Model
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Mathematical Modeling

• Simulate Results
• Run Computational Model
• to obtain Results
• draw Conclusions.
• Graphs, charts, and other visualization
• tools are useful in summarizing results
• and drawing conclusions.

Results
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Mathematical Modeling

• Interpret
• Compare conclusions with
• behavior of the real world
• problem
• If disagreement, modify Simplified
• Model and/or Mathematical model

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Syllabus

• Introduction to Matlab
• The tools of the trade
• Data analysis
• Data modeling
• Clustering
• Fourier analysis
• Simulations (Monte Carlo)

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References
Cleve Moler, Numerical Computing with MATLAB,
2004. (http//www.mathworks.com/moler)